Math, Madison, food, the Orioles, books, my kids.

Spec is representable

Saw Matt Baker at the Joint Meetings and he told me about this crazy paper he just posted, “Matroids over Hyperfields.” A hyperring is just like a ring except addition is multivalued; given elements x and y of R, x+y is a subset of R which you can think of as “the possible outcomes of summing x and y.”A hyperfield is a hyperring in which every nonzero element has a multiplicative inverse.

Here’s an example familiar to tropical geometers:let T be the hyperfield whose elements are , whose multiplication law is real addition, and whose addition law is

a + b = max(a,b) if a <> b

a + b = {c: c < a} if a=b

In other words, each element of T can be thought of as the valuation of an otherwise unspecified element of a field with a non-archimedean valuation, and then the addition law answers the question “what is ord(x+y) if ord(x) = a and ord(y) = b”?

This may sounds at first like an almost aggressively useless generalization, but no! The main point of Matt’s paper is that it makes sense to talk about a matroid with coefficients in a hyperfield, and that lots of well-studied flavors of matroids can be written as “matroids over F” for a suitable hyperfield F; in this way, a lot of different stories about different matroid theories get unified and cleaned up.

In fact, a matroid itself turns out to be the same thing as a matroid over K, where K is the Krasner hyperfield:just two elements 0 and 1, with the multiplication law you expect, and addition given by

0 + 0 = 0

0 + 1 = 1

1 + 1 = {0,1}

One thing I like about K is that it repairs the problem (if you see it as a problem) that the category of fields has no terminal object.K is terminal in the category of hyperfields; any hyperfield (and in particular any field) has a unique map to K which sends 0 to 0 and everything else to 1.

More generally, as Matt observes, if R is a commutative ring, a homomorphism f from R to K is nothing other than a prime ideal of R — namely, f^{-1}(0).So once you relax a little and accept the category of hyperfield, the functor Spec: Rings -> Sets is representable!I enjoy that.

Update: David Goss points out that this observation about Spec and the Krasner hyperfield is due to Connes and Consani in “The hyperring of adèle classes” JNT 131, (2011) 159-194, p.161. In fact, for any scheme X of finite type over Z, the underlying Zariski set of X is naturally identified with Hom(Spec(K),X); so Spec(K) functions as a kind of generic point that’s agnostic to characteristic.

7 thoughts on “Spec is representable”

Indeed. See this post by Peter Cameron: https://cameroncounts.wordpress.com/2011/07/20/the-field-with-one-element/ where he notes that “The Krasner hyperfield has elements 0 and 1: hyperaddition is given by 1+1 = {0,1} (all other instances follow from the axioms), and multiplication is obvious. This is the algebraic object which might play the role of the 1-element field.”